01418nas a2200133   4500008004100000245010100041210006900142260003500211520074700246653011900993100002101112700002101133856013001154       2014                        en  d00aTopological Invariants of Eigenvalue Intersections and Decrease of Wannier Functions in Graphene0 aTopological Invariants of Eigenvalue Intersections and Decrease   bJournal of Statistical Physics3 a<p>We investigate the asymptotic decrease of the Wannier functions for the valence and conduction band of graphene, both in the monolayer and the multilayer case. Since the decrease of the Wannier functions is characterised by the structure of the Bloch eigenspaces around the Dirac points, we introduce a geometric invariant of the family of eigenspaces, baptised eigenspace vorticity. We compare it with the pseudospin winding number. For every value n∈Z of the eigenspace vorticity, we exhibit a canonical model for the local topology of the eigenspaces. With the help of these canonical models, we show that the single band Wannier function w satisfies |w(x)|≤const |x|^{−2} as |x|→∞, both in monolayer and bilayer graphene.</p>10aWannier functions, Bloch bundles, conical intersections, eigenspace vorticity, pseudospin winding number, graphene1 aMonaco, Domenico1 aPanati, Gianluca  uhttps://www.math.sissa.it/publication/topological-invariants-eigenvalue-intersections-and-decrease-wannier-functions-graphene